Wilsonian and Functional Renormalization
Wilsonian renormalization changes resolution by integrating out or suppressing fluctuations, so the action itself moves through the space of symmetry-allowed interactions. Functional renormalization expresses that motion as an exact differential equation for an entire functional. These ideas explain why apparently irrelevant operators are generated, how fixed points organize long-distance behavior, and why a finite ansatz remains an approximation even when its parent flow equation is exact.
Enter this chapter
Section titled “Enter this chapter”Begin with Wilsonian Coarse Graining and Theory Space to see how changing resolution moves an action through theory space.
Topics
Section titled “Topics”- Wilsonian Coarse Graining and Theory Space — Defines an effective action after removing short-distance fluctuations.
- Momentum-Shell Integration and Rescaling — Combines shell integration with rescaling to obtain a flow.
- The Polchinski Exact RG Equation — Derives a differential identity for the Wilsonian interaction.
- Effective Average Actions and the Wetterich Equation — Constructs the infrared-regulated one-particle-irreducible functional.
- Functional-RG Truncations and Projection Methods — Separates the ansatz, projection, and closure choices of a calculation.
- Symmetry, Regulator Dependence, and Functional-RG Error Control — Tests symmetry restoration and the stability of approximate flows.
How the ideas fit together
Section titled “How the ideas fit together”The Polchinski and Wetterich equations evolve different functionals, but both retain infinitely many interactions in their exact forms. A finite polynomial or derivative ansatz replaces this infinite problem by a projection and a closure assumption. Consequently, an exact parent equation does not make its numerical solution exact. Enlarging the retained interactions tests one approximation; varying the regulator tests another. Symmetry identities provide additional constraints, and small regulator variation alone cannot bound all omitted effects. Dupuis et al. 2021, § 2.
Where to go next
Section titled “Where to go next”Continue to Fixed Points, Universality, and Continuum Limits to connect the eigenvectors of a linearized flow to departures from scale invariance.